text: In one spatial dimension x, for an Itô process driven by the standard Wiener process  and described by the stochastic differential equation (SDE)
  with drift  and diffusion coefficient , the Fokker–Planck equation for the probability density  of the random variable  is 
  In the following, use .
  Define the infinitesimal generator  (the following can be found in Ref.):
  The transition probability , the probability of going from  to , is introduced here; the expectation can be written as
  Now we replace in the definition of , multiply by  and integrate over . The limit is taken on
  Note now that
  which is the Chapman–Kolmogorov theorem. Changing the dummy variable  to , one gets
  which is a time derivative. Finally we arrive to
  From here, the Kolmogorov backward equation can be deduced. If we instead use the adjoint operator of , , defined such that
  then we arrive to the Kolmogorov forward equation, or Fokker–Planck equation, which, simplifying the notation , in its differential form reads
  Remains the issue of defining explicitly . This can be done taking the expectation from the integral form of the Itô's lemma:
  The part that depends on  vanished because of the martingale property.
  Then, for a particle subject to an Itô equation, using
  it can be easily calculated, using integration by parts, that
  which bring us to the Fokker–Planck equation:
  While the Fokker–Planck equation is used with problems where the initial distribution is known, if the problem is to know the distribution at previous times, the Feynman–Kac formula can be used, which is a consequence of the Kolmogorov backward equation.
  The stochastic process defined above in the Itô sense can be rewritten within the Stratonovich convention as a Stratonovich SDE:
  It includes an added noise-induced drift term due to diffusion gradient effects if the noise is state-dependent. This convention is more often used in physical applications. Indeed, it is well known that any solution to the Stratonovich SDE is a solution to the Itô SDE.
  The zero-drift equation with constant diffusion can be considered as a model of classical Brownian motion:
  This model has discrete spectrum of solutions if the condition of fixed boundaries is added for :
  It has been shown that in this case an analytical spectrum of solutions allows deriving a local uncertainty relation for the coordinate-velocity phase volume:
  Here  is a minimal value of a corresponding diffusion spectrum , while  and  represent the uncertainty of coordinate–velocity definition.